Finite Group Databasebeta

← #22571908

Subgroups — #22571908

Order 8576.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D67C212yesmaximal, complemented
2#16864061C211yesmaximal, characteristic, complemented
3D4×D268C211yesmaximal, characteristic
4#16863770C211yesmaximal, characteristic, complemented
5#16863628C212yesmaximal, complemented
6#16863597C211yesmaximal, characteristic, complemented
7#16863545C212yesmaximal, complemented
8#16863536C212yesmaximal, complemented
9C4×#11761616C211yesmaximal, characteristic, complemented
10#16863461C211yesmaximal, characteristic, complemented
11#461×C67C211yesmaximal, characteristic, complemented
12C2×Q8×D67C2212yescomplemented
13C22×D268—21no—
14C2×D4×D67C2212yes—
15C2×#11761617C2212yescomplemented
16C2×#11761616C2211yescharacteristic
17C2×#11761615C2212yescomplemented
18C2×#11761615C2212yescomplemented
19SD16×D67—28no—
20C2×C8×D67C2212yescomplemented
21#14524042—21no—
22#14524040C2211yescharacteristic
23#14524021—22no—
24#14524019C2212yescomplemented
25#14524018C2212yes—
26#14524017—21no—
27#14524013—22no—
28C4×Dic134C2211yescharacteristic, complemented
29C4×D268C2211yescharacteristic
30#14524005—21no—
31#14523974C2211yescharacteristic, complemented
32#14523966C2211yescharacteristic, complemented
33#14523965C2212yescomplemented
34#14523964C2211yescharacteristic
35(C2×D67)⋊C8C2212yescomplemented
36C4×C67⋊C8C2211yescharacteristic, complemented
37C2×SD16×C67C2212yes—
38C4⋊Q8×C67C2211yescharacteristic, complemented
39C4×D4×C67C2211yescharacteristic
40C4⋊C8×C67C2211yescharacteristic, complemented
41#103×C67C2212yescomplemented
42C23×D67—22no—
43C2×Dic134—22no—
44C2×Dic134C2311yescharacteristic
45Q8×D67—24no—
46Q8×D67—42no—
47C2×D268—22no—
48C2×D268—21no—
49C2×D268—21no—
50C2×D268—44no—
51C2×D268D411yescharacteristic
52C2×D268C2312yes—
53C2×C22⋊D67—22no—
54D4×D67—24no—
55D4×D67—42no—
56C2×C4×D67—22no—
57C2×C4×D67C2312yes—
58#11761617—24no—
59#11761616D414yescomplemented
60#11761615—24no—
61#11761615—42no—
62C2×C67⋊C8C2311yescharacteristic
63C2×C67⋊C8C2311yescharacteristic
64C8×D67—24no—
65C4⋊Dic67D411yescharacteristic
66C4⋊Dic67C2311yescharacteristic
67Dic67⋊C4—21no—
68Dic67⋊C4—22no—
69(C2×D67)⋊C4—22no—
70(C2×D67)⋊C4—22no—
71(C2×D67)⋊C4D412yes—
72C4×Dic67—21no—
73C2×Q8×C67C2312yes—
74C2×D4×C67C2311yescharacteristic
75C22×C4×C67—21no—
76SD16×C67—24no—
77C2×C8×C67C2312yes—
78C4⋊C4×C67—21no—
79C4⋊C4×C67—21no—
80C4⋊C4×C67C2311yescharacteristic
81C22⋊C4×C67—21no—
82C42×C67C2311yescharacteristic
83C23×C67—21no—
84C22×D67—22no—
85C22×D67—44no—
86C22×D67—44no—
87C22×D67—82no—
88C22×D67C2×D412yes—
89Q8×C67—22no—
90Q8×C67—22no—
91D4×C67—21no—
92D4×C67C2×D412yes—
93C2×C4×C67—21no—
94C2×C4×C67—22no—
95C2×C4×C67—42no—
96C2×C4×C67C2×D411yescharacteristic
97C2×C4×C67C2×D411yescharacteristic
98C2×C4×C67C2411yescharacteristic
99C22⋊D67—44no—
100C2×Dic67—21no—
101C2×Dic67—21no—
102C2×Dic67C2×D412yes—
103D268—22no—
104D268—22no—
105D268—44no—
106D268—42no—
107D268—81no—
108D268SD1614yescomplemented
109C4×D67—22no—
110C4×D67—22no—
111C4×D67—44no—
112Dic134—21no—
113Dic134—44no—
114Dic134C2×D412yescomplemented
115C8×C67—22no—
116C67⋊C8—21no—
117C67⋊C8C2×D412yescomplemented
118C22×C67—22no—
119C22×C67—41no—
120C22×C67—41no—
121C22×C67C22×D411yescharacteristic
122C2×D67—22no—
123C2×D67—22no—
124C2×D67—42no—
125C2×D67—42no—
126C2×D67—42no—
127C2×D67—88no—
128C2×D67C2×SD1614yescomplemented
129C4×C67—21no—
130C4×C67—42no—
131C4×C67—41no—
132C4×C67C2×SD1612yescomplemented
133C4×C67C22×D411yescharacteristic
134C4×C67H(5,𝔽2)11yescharacteristic
135Dic67—22no—
136Dic67—22no—
137Dic67—41no—
138C2×C67—42no—
139C2×C67D4211yescharacteristic
140C2×C67C22×SD1611yescharacteristic
141C2×C67#57711yescharacteristic
142D67—24no—
143D67—82no—
144D4×SD16—671nomaximal, Sylow
145C67D4×SD1611yescharacteristic, complemented, Sylow
146C22×SD16—672no—
147D4×Q8—671no—
148D42—671no—
149#492—671no—
150#465—672no—
151#461D6711yescharacteristic, complemented
152#460—671no—
153#450—672no—
154#448—672no—
155C4×SD16—671no—
156C8×D4—671no—
157C22×Q8—672no—
158C22×D4—672no—
159C22×D4—1341no—
160C2×SD16—672no—
161C2×SD16—671no—
162C2×SD16—672no—
163C2×SD16—672no—
164C2×SD16—1348no—
165C2×SD16C2×D6712yes—
166C22×C8—672no—
167C4⋊Q8—1341no—
168C4⋊Q8C2×D6711yescharacteristic, complemented
169#127—671no—
170C22⋊Q8—672no—
171C22⋊Q8—1342no—
172#121—672no—
173#121—1341no—
174#120—1342no—
175C4×Q8—671no—
176C4×D4—671no—
177C4×D4—1341no—
178C4×D4C2×D6711yescharacteristic
179#106—671no—
180C4⋊C8C2×D6711yescharacteristic, complemented
181#103—671no—
182#103C2×D6712yescomplemented
183#102—671no—
184#102—672no—
185C22⋊C8—672no—
186C4×C8—671no—
187C24—1342no—
188C2×Q8—2682no—
189C2×Q8—671no—
190C2×Q8—1342no—
191C2×Q8—1344no—
192C2×Q8C22×D6712yes—
193C2×D4—2684no—
194C2×D4—2682no—
195C2×D4—671no—
196C2×D4—672no—
197C2×D4—1342no—
198C2×D4—1342no—
199C2×D4—1344no—
200C2×D4—1341no—
201C2×D4—1341no—
202C2×D4C22×D6711yescharacteristic
203C22×C4—21no—
204C22×C4—672no—
205C22×C4—1342no—
206SD16—24no—
207SD16—2682no—
208SD16—674no—
209SD16—1344no—
210SD16—1344no—
211C2×C8—671no—
212C2×C8—671no—
213C2×C8—1344no—
214C2×C8C22×D6712yes—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—671no—
218C4⋊C4—671no—
219C4⋊C4—1342no—
220C4⋊C4—1341no—
221C4⋊C4C22×D6711yescharacteristic
222C22⋊C4—21no—
223C22⋊C4—672no—
224C22⋊C4—1342no—
225C22⋊C4—1342no—
226C42—1341no—
227C42C22×D6711yescharacteristic
228C23—21no—
229C23—2684no—
230C23—2684no—
231C23—5362no—
232C23—672no—
233C23—1342no—
234Q8—22no—
235Q8—22no—
236Q8—2684no—
237Q8—672no—
238Q8—1341no—
239D4—21no—
240D4—2684no—
241D4—2684no—
242D4—2682no—
243D4—5361no—
244D4—674no—
245D4—1342no—
246D4—1342no—
247D4D4×D6712yes—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—42no—
251C2×C4—2684no—
252C2×C4—672no—
253C2×C4—1341no—
254C2×C4—1341no—
255C2×C4—1342no—
256C2×C4—1342no—
257C2×C4D4×D6711yescharacteristic
258C2×C4D4×D6711yescharacteristic
259C2×C4C23×D6711yescharacteristic
260C8—22no—
261C8—672no—
262C8—1341no—
263C22—22no—
264C22—41no—
265C22—41no—
266C22—2682no—
267C22—2682no—
268C22—2682no—
269C22—5368no—
270C22—674no—
271C22—1342no—
272C22—1342no—
273C22C2×D4×D6711yescharacteristic
274C4—21no—
275C4—42no—
276C4—41no—
277C4—2681no—
278C4—1342no—
279C4—1342no—
280C4SD16×D6712yescomplemented
281C4C2×D4×D6711yescharacteristic
282C4#1452408911yescharacteristic
283C2—42no—
284C2—5362no—
285C2—1344no—
286C2#1686403011yescharacteristic
287C2C2×SD16×D6711yescharacteristic
288C2#1686420511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#461×C67C2
socleC22×C67C22×D4
hypercenter#461D67
hypocenterC67D4×SD16
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×SD16×D67C2semidirect
2. #16864061C2semidirect
4. #16863770C2semidirect
5. #16863628C2semidirect
6. #16863597C2semidirect
7. #16863545C2semidirect
8. #16863536C2semidirect
9. C4×#11761616C2semidirect
10. #16863461C2semidirect
11. #461×C67C2semidirect
12. C2×Q8×D67C22semidirect
15. C2×#11761617C22semidirect
17. C2×#11761615C22semidirect
18. C2×#11761615C22semidirect
20. C2×C8×D67C22semidirect
24. #14524019C22semidirect
28. C4×Dic134C22semidirect
31. #14523974C22semidirect
32. #14523966C22semidirect
33. #14523965C22semidirect
35. (C2×D67)⋊C8C22semidirect
36. C4×C67⋊C8C22semidirect
38. C4⋊Q8×C67C22semidirect
40. C4⋊C8×C67C22semidirect
41. #103×C67C22semidirect
59. #11761616D4semidirect
108. D268SD16semidirect
114. Dic134C2×D4semidirect
117. C67⋊C8C2×D4semidirect
128. C2×D67C2×SD16semidirect
132. C4×C67C2×SD16semidirect
145. C67D4×SD16semidirect
151. #461D67semidirect
168. C4⋊Q8C2×D67semidirect
180. C4⋊C8C2×D67semidirect
182. #103C2×D67semidirect
280. C4SD16×D67semidirect